Symbolic iterative method for forward solution of geodetic line in long-wave and very long-wave positioning

Based on the geometric definitions of geodetic line and the Bessel projection on an ellipsoid, this paper establishes a unified notation system to resolve previous inconsistencies. Using computer algebra and the concept of symbolic iteration, a direct expression for the forward solution of the geodetic line is derived. This approach offers a simplified, non-iterative representation of the geodetic line, eliminating the need for repeated numerical iterations in subsequent calculations. Consequently, three equivalent formulations of the exact direct solution are presented. Numerical experiments show that, compared to conventional iterative methods, the computational error in long-distance geodetic lines remains below 0.6 m, meeting the accuracy demands of high-precision applications. Validation using the long-wave navigation stations in Dunhuang, Naqu, and Korla further demonstrates the practicality of the proposed algorithm, as evidenced by verifying the exact solution for a 1 000 km geodetic line.